Nuttun nearby.
If you go by time rather than distance, HPP is probably closest.
There's a few weirs around Warwick and Stratford that are okay with water for a splash around.
Mattlock is probably next closest but a bit crap.
Otherwise you are into Wales.

You need high hills for grade 3 or above. If you look at a relief map of Britain, you'll see that the area around Birmingham is not very hilly.

Shropshire and the Peak District are hilly regions that are closer to Birmingham than Wales, but for some reason neither has good white water, as far as I know.

Living in East Anglia (even flatter and further from hills than Birmingham), I just accept that 80% of my paddling is going to be at Lee Valley.

Peak District has a few good runs, but you need to see zebras queueing up in twos before they're runnable

EDIT - a bit more info:

Runs that occasionally get when it tips down - Wye (Buxton to Litton) G2(3), Derwent (Bamford to Calver) G2(3), Ashop G2-4 depending on who you ask, where they got in and how high the water was), Etherow 3-4 as above, River Noe G2(3), River Dane G2(3), Heyden Brook (4/5??? can't remember)

Also rumoured that Burbage Brook has been done a couple of times at a Grade 2/3 ish tree strewn pinball experience, and it's continuation down Padley Gorge at a death defying grade 5 ish.

The upper Don, from Deepcar to Oughtibridge North of Sheffield deserves a mention at Grade 2/3 ish too. It's probably easier to catch than any of the true spate streams, but even so you only really get a few hours after the rain stops.

Also rumoured that Burbage Brook has been done a couple of times at a Grade 2/3 ish tree strewn pinball experience, and it's continuation down Padley Gorge at a death defying grade 5 ish.

I saw Bar Brook in a similar condition one day last winter, the bit that runs parallel to the A621 as you drive from Sheffield down in to Baslow. Anyone surviving the experience would have been deposited into the Derwent on the Chatsworth Estate, where they'd have been over the two big weirs and away towards Matlock before anyone could stop them. ;-)

The upper Don, from Deepcar to Oughtibridge North of Sheffield deserves a mention at Grade 2/3 ish too. It's probably easier to catch than any of the true spate streams, but even so you only really get a few hours after the rain stops.

Good shout - I'd forgotten about that - and really ought to do it soon!

I saw Bar Brook in a similar condition one day last winter, the bit that runs parallel to the A621 as you drive from Sheffield down in to Baslow. Anyone surviving the experience would have been deposited into the Derwent on the Chatsworth Estate, where they'd have been over the two big weirs and away towards Matlock before anyone could stop them. ;-)